Fourth order differential operators with distributional potentials

Abstract

In this paper, regular and singular fourth order differential operators with distributional potentials areinvestigated. In particular, existence and uniqueness of solutions of the fourth order differential equations are proved,deficiency indices theory of the corresponding minimal symmetric operators are studied. These symmetric operators areconsidered as acting on the single and direct sum Hilbert spaces. The latter one consists of three Hilbert spaces such thata squarely integrable space and two spaces of complex numbers. Moreover all maximal self-adjoint, maximal dissipativeand maximal accumulative extensions of the minimal symmetric operators including direct sum operators are given inthe single and direct sum Hilbert spaces.